Keeping plane 1-sided while UVing
In your properties panel, it looks like its a cube, not a plane. I can also tell because in your UV, there are more than 2 faces. There's actually 6.
To fix it, just edit your UVs. It won't take long at all. Or you can completely restart your UV job if you wish. I recommend doing so with one of the following methods.
Regular Unwrapping
Go into edit mode, sleect the faces you want to be in your UV, and hit U and then click on unwrap. You can edit the UVs further from the UV editor, as I am sure you are aware.
Project From View
If you don't already know how to project from view, I suggest reading up on it as it is quite useful and I use it a lot. In your case, it should get the job done instantly.